Haag's theorem establishes that free fields and interacting fields exist in different, unitarily inequivalent Hilbert spaces, meaning one cannot transform into the other. The theorem highlights the mathematical ill-definition of transitioning from a free to an interacting field due to infrared divergences. For a deeper understanding, refer to A. Duncan's book, "The Conceptual Framework of Quantum Field Theory," specifically section 10.5. Alternative approaches, such as the algebraic approach to quantum field theory (QFT), effectively address the challenges posed by Haag's theorem.
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That's not quite what Haag's theorem says. A free field and an interacting field are different things, and one cannot "become" the other, regardless of what Haag's theorem or any other mathematical result says.lindberg said:according to Haag's theorem, a free field cannot become an interacting one?
Wasn't Haag's conclusion extended later to other approaches?PeterDonis said:To someone who is used to the usual way of modeling interacting fields as perturbations of free fields, this seems like a problem; but there are other approaches to quantum field theory, such as the algebraic approach, in which it is not a problem at all.
Given that the whole point of the algebraic approach to QFT is to be able to deal with unitarily inequivalent representations, showing that the algebraic approach leads to unitarily inequivalent representations isn't much of an issue.lindberg said:Wasn't Haag's conclusion extended later to other approaches?